Null angular momentum and weak KAM solutions of the Newtonian \(N\)-body problem. (English) Zbl 1383.37050
This paper presents an extension of the result given by B. Percino and H. Sánchez-Morgado [Arch. Ration. Mech. Anal. 213, No. 3, 981–991 (2014; Zbl 1342.70031)], where it was shown that the Busemann function of the parabolic homothetic motion for a minimal central configuration of the \(N\)-body problem is a viscosity solution of the Hamilton-Jacobi equation.
The author takes the same hypotheses by the aforementioned article and uses weak KAM solutions of the Hamiltonian-Jacobi equations [G. Contreras, Calc. Var. Partial Differ. Equ. 13, No. 4, 427–458 (2001; Zbl 0993.37030)] to prove the existence of completely parabolic motions. The key mechanism is to give conditions so that a weak KAM solution is rotational invariant. The result is achieved by studying the angular momentum for the calibrating curves of rotational invariant solutions and characterizing invariant solutions as those where calibrating curves have zero angular momentum.
The author takes the same hypotheses by the aforementioned article and uses weak KAM solutions of the Hamiltonian-Jacobi equations [G. Contreras, Calc. Var. Partial Differ. Equ. 13, No. 4, 427–458 (2001; Zbl 0993.37030)] to prove the existence of completely parabolic motions. The key mechanism is to give conditions so that a weak KAM solution is rotational invariant. The result is achieved by studying the angular momentum for the calibrating curves of rotational invariant solutions and characterizing invariant solutions as those where calibrating curves have zero angular momentum.
Reviewer: Martha Alvarez-Ramirez (México City)
MSC:
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 37N05 | Dynamical systems in classical and celestial mechanics |
| 37J15 | Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) |
| 37J50 | Action-minimizing orbits and measures (MSC2010) |
| 70F10 | \(n\)-body problems |
| 70H20 | Hamilton-Jacobi equations in mechanics |
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